HW2_6337_2023
2023-03-23
Authors : Mankirat Singh Bhamra MXB220061 - Harikrishna Dev HXD220000 - Sarthak Vajpayee SXV220020 - Krishnan Venkatesan KXV220007
Loading required libraries and cleaning environment
rm(list = ls())
demo = T
require(psych)## Loading required package: psychrequire(data.table)## Loading required package: data.tablerequire(dplyr)## Loading required package: dplyr##
## Attaching package: 'dplyr'## The following objects are masked from 'package:data.table':
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## between, first, last## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionrequire(ggplot2)## Loading required package: ggplot2##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
##
## %+%, alpharequire(rlist)## Loading required package: rlistif(demo) {setwd("~/Library/Mobile Documents/com~apple~CloudDocs/School Work/Sem 2/BUAN 6337/HW/pset3")}1 (R) (Weak) Law of Large Numbers • (a) Consider a continuous random variable Xi ∼ Uni f orm [0, 2]. What is E [Xi] and Var(Xi)?
Ans: The mean (expected value) of a continuous uniform distribution [a, b] is given by:
\[ E[X] = \frac{(a + b)}{2} \]
In this case, a = 0 and b = 2, so we have:
\[ E[Xi] = \frac{(0 + 2)} {2}= 1 \]
Therefore, the expected value of Xi is 1.
The variance of a continuous uniform distribution [a, b] is given by:
\[ Var[X] = \frac{(b - a)^2}{12} \]
In this case, a = 0 and b = 2, so we have:
\[ Var[Xi] = \frac{(2 - 0)^2}{12}= \frac{1}{3} \]
Therefore, the variance of Xi is \(\frac{1}{3}\).
• (b) Consider the Xi defined above in (a) for i = 1, 2, …, n, where each Xi ⊥ Xj whenever i ̸= j. Consider the sample mean
What is E [X ̄ n] and Var (X ̄ n)? (The answer might be a function of n).
Ans: The sample mean of n independent and identically distributed (i.i.d.) random variables Xi has the following properties:
\[ E[X̄n] = E[\frac{1}{n} ∑X_i] =\frac{1}{n} ∑E[X_i] =\frac{1}{n} \times n\times E[X_i] = E[X_i]\]
\[Var[X̄n] = Var[\frac{1} {n} ∑X_i] =\frac{1}{n^2} ∑Var[X_i]\]
Since each Xi is uniformly distributed on [0, 2], we know from part (a) that \(E[Xi] = 1\) and \(Var[Xi] = \frac{1}{3}\). Therefore:
\[E[X̄n] = E[Xi] = 1\]
\[ Var[X̄n] = Var[\frac {1} {n} ∑X_i] =\frac{1}{n^2} ∑Var[X_i] = \frac{1}{n^2}\times n\times\frac{1} {3}= \frac{1} {3n}\]
Therefore, the expected value of the sample mean X̄n is 1, and the variance of the sample mean X̄n is \(\frac{1}{3n}\).
Now repeat the following (c)-(e) for \(n = 1, 2, 3, 5, 10, 50, 100, 1000, 3000\). Use the for loops to execute (c)-(e). • (c) (R) Generate a size n vector of independent Uni f orm [0, 2] random variables and cal- culate its sample mean X ̄ n. • (d)(R)Take \(|X ̄n−E[Xi]|\) and report the value. • (e) (R) Now consider a continuous transformation $ f(x) = 2x^2 − 5x + 1 + 1 $. Take \(| f ( X ̄ n ) − f ( E [ X i ] ) |\)
n = c(1, 2, 3, 5, 10, 50, 100, 1000, 3000)
Q1 = function(n){
# c
x = runif(n,0,2)
x_bar = mean(x)
#d
e_x = 1
diff = abs(x_bar - e_x)
#e
quad = function(x)
{
return (2*x^2 - 5*x + 1 + (1/(3*x)))
}
diff_function = abs(quad(x_bar)-quad(e_x))
return(list(n=n,mean=x_bar,diff=diff,diff_func=diff_function))
}
for(i in n){
assign(paste0("v",i),Q1(i))
if(demo){print(get(paste0("v",i)))}
}## $n
## [1] 1
##
## $mean
## [1] 0.3231701
##
## $diff
## [1] 0.6768299
##
## $diff_func
## [1] 2.291143
##
## $n
## [1] 2
##
## $mean
## [1] 1.162018
##
## $diff
## [1] 0.1620183
##
## $diff_func
## [1] 0.1559946
##
## $n
## [1] 3
##
## $mean
## [1] 1.572479
##
## $diff
## [1] 0.572479
##
## $diff_func
## [1] 0.03836837
##
## $n
## [1] 5
##
## $mean
## [1] 1.55405
##
## $diff
## [1] 0.5540496
##
## $diff_func
## [1] 0.05894767
##
## $n
## [1] 10
##
## $mean
## [1] 1.016361
##
## $diff
## [1] 0.01636073
##
## $diff_func
## [1] 0.02119117
##
## $n
## [1] 50
##
## $mean
## [1] 1.017167
##
## $diff
## [1] 0.01716702
##
## $diff_func
## [1] 0.02220337
##
## $n
## [1] 100
##
## $mean
## [1] 0.9505482
##
## $diff
## [1] 0.04945179
##
## $diff_func
## [1] 0.07168425
##
## $n
## [1] 1000
##
## $mean
## [1] 1.007704
##
## $diff
## [1] 0.007703591
##
## $diff_func
## [1] 0.01013313
##
## $n
## [1] 3000
##
## $mean
## [1] 0.9947288
##
## $diff
## [1] 0.00527115
##
## $diff_func
## [1] 0.007093081• (f) What happens to the reported value in (d) and (e) as n increases? Discuss.
Ans: As n increases, the difference between the theoretical value and practical value tends towards 0. This implies larger the sample, the results are better statistically.
diff_func = c()
for(i in n){
diff_func = append((Q1(i)$diff_func),diff_func)
}
plot(rev(diff_func))
2 (R) The Central Limit Theorem • (a) Consider a continuous random variable \(Xi ∼ Uniform [0, 2]\). What is \(E[X_i]\) and \(Var(X_i)\)?
\[ Mean = \frac {a+b}{2} = 1 \] \[ Var = \frac {1}{3} \]
• (b) Consider the Xi defined above in (a) for i = 1, 2, …, n, where each \(Xi ⊥ Xj\) whenever \(i \ne j\). Consider the sample mean ̄1n Xn := n ∑Xi. i=1 What is \(E[X ̄ n]\) and \(Var(X ̄ n)\)? (The answer would be a function of n).
Ans:
\[ E[X_n] = \frac{1}{n} \times {n} \times 1 = 1 \] \[ E[X_{n=6}] = 1 \]
\[ Var[X_n] = \frac{1}{\sqrt{n}} \times n \times 0.3333 = 0.041667 \]
• (c) Consider the transformation \[ Y_n : = \sqrt n \times ( X _n − E[ X_i ] ) \] What is E [Yn] and Var (Yn)?
Ans:
\[E[Y_n] = \sqrt {n} \times 0 = 0 \]
\[ var = \frac {1}{3} \]
• (d) Consider the transformation √ ( X ̄ n − E [ X i ] ) Zn:= npVar(Xi). What is E [Zn] and Var (Zn)?
\[ E[Z_n] = 0 \]
\[ Var[Z_n] = 1 \]
Now repeat the following for n = 1, 2, 3, 5, 10, 50, 100, 1000, 3000. Use the for loops to execute (e)-(m). • (e)(R)Generate t = 1,2,3,4,5,6,7,8,9,10,11,…,2500 n×1 vectors of independent \(Uniform[0,2]\) random variables and calculate its sample mean \(X ̄ nt\) respectively for each t. Denote this size 2500 vector as \[v_2500 ≡X ̄_1,X ̄_2,...,X ̄_2500\] for now.
n = c(1, 2, 3, 5, 10, 50, 100, 1000, 3000)
Q2 = function(n){
t = c(1:2500)
v = replicate(length(t), runif(i, 0, 2))
if (n == 1) {
vn_2500 = v
} else {
vn_2500 = apply(v, 2, mean)
}
vn_2500_bar = mean(vn_2500)
vn_2500_var = var(vn_2500)
e_xi = 1
yn_2500 = sqrt(n)*(vn_2500 - e_xi)
mean_yn_2500 = mean(yn_2500)
var_yn_2500 = var(yn_2500)
var_xi = 1/3
zn_2500 = sqrt(n)*(vn_2500 - e_xi)/sqrt(var_xi)
mean_zn_2500 = mean(zn_2500)
var_zn_2500 = var(zn_2500)
return(list(n=n, vn = vn_2500, mean_vn = vn_2500_bar, var_vn = vn_2500_var, zn = zn_2500, mean_zn = mean_zn_2500,yn = yn_2500, var_zn = var_zn_2500))
}
for(i in n){
assign(paste0("f",i),Q2(i))
}
for(i in n){
hist(Q2(i)$zn, breaks = 100,
main = paste("Histogram of zn_2500 for n = ", i, sep = ""),
xlab = "zn_t values",
xlim = c(-3,3))
}
for(i in n){
hist(Q2(i)$yn, breaks = 100,
main = paste("Histogram of zn_2500 for n = ", i, sep = ""),
xlab = "vn_t values",
xlim = c(-3,3))
}
• (m) What happens to the reported values in (f), (h), (k) and histograms in (i) and (l) as n increases? Discuss.
3 (R) WLLN with Simple Regression
In this exercise, you will generate datasets for simple regression yourself, and then try to esti- mate the model parmaters to examine the properties of simple regression OLS estimators as the sample size n grows. Repeat the following for n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 50, 75, 100, 250, 500, 1000, 2000, 3000. Use the for loops as necessary.
• (a) (R) Generate a n × 1 column vector of Uniform[0, 12] random variable and denote it as x.
• (b) (R) Generate a n × 1 column vector of Uni f orm [−4, 4] random variable and denote it as u. • (c) (R) Generate the y vector using the following formula: yi = 3 + 2xi + ui for each i = 1, 2, 3, …, n. That is, i’th row of x and u corresponds to i’th observation. • (d) (R) Now you have a Monte-Carlo dataset of size n. Estimate the β in the following model using OLS. (Recall the formula d .) What is the calculated value of βOLS,n? Report.
n = c(2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 50, 75, 100, 250, 500, 1000, 2000, 3000)
Q3 <- function(n) {
x = matrix(runif(n, 0, 12), nrow = i, ncol = 1)
u = matrix(runif(n, -4, 4), nrow = i, ncol = 1)
y = 3 + (2 * x) + u
cov_xy = cov(x, y)
var_x = var(x)
hat_beta = cov_xy/var_x
abs_beta = abs(hat_beta - 2)
return(list(n=n,x=x,u=u,y=y,cov_xy=cov_xy,var_x=var_x,hat_beta=hat_beta,abs_beta=abs_beta))
}
for(i in n){
assign(paste0("x",i),Q3(i))
}4 In this exercise, you will generate datasets for simple regression yourself, and then try to esti- mate the model parmaters to examine the properties of simple regression OLS estimators as the sample size n grows. Repeat the following for n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 50, 75, 100, 250, 500, 1000, 2000, 3000. Use the for loops as necessary.
• (a) (R) Generate a n × 1 column vector of Uni f orm [0, 12] random variable and denote it as x. • (b) (R) Generate a n × 1 column vector of Uni f orm [−4, 4] random variable and denote it as u. • (c) (R) Generate the y vector using the following formula: yi = 3 + 2xi + ui for each i = 1, 2, 3, …, n. That is, i’th row of x and u corresponds to i’th observation. • (d) (R) Now you have a Monte-Carlo dataset of size n. Estimate the β in the following model \[yi = α + βxi + ui \]
• (e) (R) Repeat (a)-(d) for 2,500 times. You must have 2,500 βˆOLS,n estimates in the memory ˆ1 ˆ2 ˆ2500 ′ at the end of this sub-question. Denote this size 2,500 vector by b = βOLS,n, βOLS,n, …, βOLS,n . • (f) (R) Calculate the variance of b and report. • (g) (R) Subtract 2 from b and multiply (c1, c2, …., c2500), i.e., n on each element of b and denote this as \[ c_i = \sqrt{n} \times (\hat \beta ^ i _{OLS,n} - 2)\] using OLS. Save it in the memory.
n = c(2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 50, 75, 100, 250, 500, 1000, 2000, 3000)
Q4 <- function(n){
b <- c()
for (j in c(1:2500)) {
# a
x = matrix(runif(n, 0, 12), nrow = n, ncol = 1)
# b
u = matrix(runif(n, -4, 4), nrow = n, ncol = 1)
# c
y = 3 + (2 * x) + u
# d
cov_xy = cov(x, y)
var_x = var(x)
beta_hat = cov_xy/var_x
b = append(b, beta_hat)
}
# f
var_b = var(b)
c = sqrt(n)*(b - 2)
return(list(n=n,b=b,var_b=var_b,c=c))
}
for(i in n){
assign(paste0("q",i),Q4(i))
hist_data <- hist(Q4(i)$c, breaks = 100,
main = paste("Histogram of c for n = ", i, sep = ""),
xlab = "c_i values",
xlim = c(-3,3))
x_values <- seq(min(Q4(i)$c), max(Q4(i)$c), length = 100)
y_values <- dnorm(x_values, mean = mean(Q4(i)$c), sd = sd(Q4(i)$c))
y_values <- y_values * diff(hist_data$mids[1:2]) * length(Q4(i)$c)
lines(x_values, y_values, lwd = 2)
}
• (h) What happens to the reported values in (f) and the histogram in (g) as n grows large? Discuss.
- Video Game Sales Regression
The dataset for this exercise is available in VideoGamesSales_Main.csv. This dataset contains information on the global sales and critic and user review ratings for videogames launched between 2001 and 2012 (from www.vgchartz.com).
• Your task is to develop a regression model (using lm()) that links global sales to video game reviews, and explore ways in which the model fit could be improved through suit- able changes to the model specification and vdgmriables. (a) (R) First, create a frequency table of 3 variables: platform, genre, and rating.
vdgm <- fread("videogamesales_main.csv",header = T)
if(demo) {str(vdgm)}## Classes 'data.table' and 'data.frame': 4413 obs. of 12 variables:
## $ Name : chr "Madden NFL 11" "Sly Cooper and the Thievius Raccoonus" "The Lord of the Rings: Conquest" "Red Steel 2" ...
## $ Platform : chr "PS3" "PS2" "X360" "Wii" ...
## $ Genre : chr "Sports" "Platform" "Action" "Shooter" ...
## $ Publisher : chr "Electronic Arts" "Sony Computer Entertainment" "Electronic Arts" "Ubisoft" ...
## $ Developer : chr "EA Tiburon" "Sucker Punch" "Pandemic Studios" "Ubisoft Paris" ...
## $ Rating : chr "E" "E" "T" "T" ...
## $ Global_Sales : num 2.38 1.21 0.63 0.62 1.43 0.09 0.56 0.33 1.52 0.21 ...
## $ Year_of_Release: int 2010 2002 2009 2010 2010 2011 2008 2009 2002 2006 ...
## $ Critic_Score : int 83 86 55 80 61 70 78 75 38 70 ...
## $ Critic_Count : int 36 41 58 73 59 4 47 17 21 9 ...
## $ User_Score : num 6.1 8.6 7 8.6 5.7 7 7.6 7.8 5 6.5 ...
## $ User_Count : int 68 184 110 178 180 4 36 6 26 10 ...
## - attr(*, ".internal.selfref")=<externalptr>ptl_frq <- table(vdgm$Platform)
gnr_frq <- table(vdgm$Genre)
rating_frq <- table(vdgm$Rating)- Create categorial variables for platform, genre, and rating using data.table. Also create a variable for the age of the game relative to year 2013 (Note that these games were released before 2013).
vdgm_2013 <- vdgm[,Platform]
vdgm$Action = ifelse(vdgm$Genre =='Action', 1, 0)
vdgm$adventure = ifelse (vdgm$Genre == 'Adventure',1, 0)
vdgm$fighting = ifelse(vdgm$Genre =='Fighting', 1, 0)
vdgm$misc = ifelse (vdgm$Genre == 'Misc',1, 0)
vdgm$platform = ifelse(vdgm$Genre =="Platform",1, 0)
vdgm$puzzle = ifelse(vdgm$Genre =="Puzzle", 1, 0)
vdgm$racing = ifelse(vdgm$Genre =='Racing', 1, 0)
vdgm$Role_playing = ifelse(vdgm$Genre == "Role-Playing", 1, 0)
vdgm$Shooter = ifelse(vdgm$Genre =="Shooter", 1, 0)
vdgm$simulation = ifelse(vdgm$Genre == 'Simulation', 1, 0)
vdgm$sports = ifelse(vdgm$Genre == "Sports", 1, 0)
vdgm$strategy = ifelse(vdgm$Genre == 'strategy',1,0)
vdgm$Action = ifelse(vdgm$Genre =='Action', 1, 0)
vdgm$adventure = ifelse (vdgm$Genre == 'Adventure',1, 0)
vdgm$fighting = ifelse(vdgm$Genre =='Fighting', 1, 0)
vdgm$misc = ifelse (vdgm$Genre == 'Misc',1, 0)
vdgm$platform = ifelse(vdgm$Genre =="Platform",1, 0)
vdgm$puzzle = ifelse(vdgm$Genre =="Puzzle", 1, 0)
vdgm$racing = ifelse(vdgm$Genre =='Racing', 1, 0)
vdgm$Role_playing = ifelse(vdgm$Genre == "Role-Playing", 1, 0)
vdgm$Shooter = ifelse(vdgm$Genre =="Shooter", 1, 0)
vdgm$simulation = ifelse(vdgm$Genre == 'Simulation', 1, 0)
vdgm$sports = ifelse(vdgm$Genre == "Sports", 1, 0)
vdgm$strategy = ifelse(vdgm$Genre == 'strategy',1,0)
vdgm$DS = ifelse (vdgm$Platform == 'DS',1, 0)
vdgm$GBA = ifelse(vdgm$Platform =='GBA',1, 0)
vdgm$GC = ifelse(vdgm$Platform =='GC',1,0)
vdgm$Pc = ifelse(vdgm$Platform=='PC',1,0)
vdgm$Ps2 = ifelse(vdgm$Platform =="PS2",1,0)
vdgm$Ps3 = ifelse(vdgm$Platform =="PS3",1,0)
vdgm$PSP = ifelse (vdgm$Platform=="PSP",1,0)
vdgm$Wii = ifelse (vdgm$platform =="Wii",1,0)
vdgm$X360 = ifelse(vdgm$Platform =="x360",1,0)
vdgm$XB = ifelse (vdgm$Platform == "XB",1,0)
vdgm$DS = ifelse(vdgm$Platform == "DS",1,0)
vdgm$E = ifelse(vdgm$Rating == "E",1,0)
vdgm$`E10+` = ifelse(vdgm$Rating == "E10+",1,0)
vdgm$M = ifelse(vdgm$Rating =='M' , 1,0)
vdgm$T = ifelse(vdgm$Rating == "T",1,0)- Run a regression with all relevant X variables. Report the adjusted R-squared.
lm1 <- lm(Global_Sales ~. - Name - Publisher - Developer,data = vdgm)
summary(lm1)##
## Call:
## lm(formula = Global_Sales ~ . - Name - Publisher - Developer,
## data = vdgm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.931 -0.603 -0.157 0.287 79.609
##
## Coefficients: (26 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.487e+01 3.283e+01 2.280 0.022635 *
## PlatformGBA -3.655e-01 1.932e-01 -1.891 0.058656 .
## PlatformGC -5.773e-01 1.738e-01 -3.321 0.000905 ***
## PlatformPC -1.098e+00 1.564e-01 -7.020 2.57e-12 ***
## PlatformPS2 -1.714e-01 1.429e-01 -1.199 0.230416
## PlatformPS3 -2.784e-01 1.474e-01 -1.889 0.058954 .
## PlatformPSP -3.523e-01 1.574e-01 -2.238 0.025284 *
## PlatformWii 5.711e-01 1.485e-01 3.846 0.000122 ***
## PlatformX360 -5.233e-01 1.472e-01 -3.555 0.000382 ***
## PlatformXB -7.815e-01 1.597e-01 -4.893 1.03e-06 ***
## GenreAdventure -2.025e-01 1.695e-01 -1.195 0.232235
## GenreFighting 7.171e-02 1.452e-01 0.494 0.621327
## GenreMisc 3.774e-01 1.375e-01 2.744 0.006099 **
## GenrePlatform 3.196e-02 1.451e-01 0.220 0.825629
## GenrePuzzle -4.010e-01 2.479e-01 -1.618 0.105776
## GenreRacing 5.483e-02 1.237e-01 0.443 0.657609
## GenreRole-Playing -3.169e-01 1.170e-01 -2.710 0.006762 **
## GenreShooter -4.054e-02 1.068e-01 -0.380 0.704212
## GenreSimulation 2.353e-01 1.533e-01 1.534 0.125031
## GenreSports -1.156e-02 1.204e-01 -0.096 0.923500
## GenreStrategy -2.469e-01 1.635e-01 -1.510 0.131096
## RatingE10+ -3.931e-01 1.069e-01 -3.678 0.000238 ***
## RatingM -5.186e-01 1.186e-01 -4.374 1.25e-05 ***
## RatingT -4.798e-01 9.204e-02 -5.213 1.95e-07 ***
## Year_of_Release -3.745e-02 1.635e-02 -2.291 0.022035 *
## Critic_Score 1.649e-02 3.147e-03 5.241 1.67e-07 ***
## Critic_Count 2.981e-02 2.269e-03 13.139 < 2e-16 ***
## User_Score -5.491e-02 2.779e-02 -1.976 0.048261 *
## User_Count 8.748e-04 6.559e-05 13.337 < 2e-16 ***
## Action NA NA NA NA
## adventure NA NA NA NA
## fighting NA NA NA NA
## misc NA NA NA NA
## platform NA NA NA NA
## puzzle NA NA NA NA
## racing NA NA NA NA
## Role_playing NA NA NA NA
## Shooter NA NA NA NA
## simulation NA NA NA NA
## sports NA NA NA NA
## strategy NA NA NA NA
## DS NA NA NA NA
## GBA NA NA NA NA
## GC NA NA NA NA
## Pc NA NA NA NA
## Ps2 NA NA NA NA
## Ps3 NA NA NA NA
## PSP NA NA NA NA
## Wii NA NA NA NA
## X360 NA NA NA NA
## XB NA NA NA NA
## E NA NA NA NA
## `E10+` NA NA NA NA
## M NA NA NA NA
## T NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.946 on 4384 degrees of freedom
## Multiple R-squared: 0.1795, Adjusted R-squared: 0.1742
## F-statistic: 34.25 on 28 and 4384 DF, p-value: < 2.2e-16- Now, generate natural log of the following variables: global sales, critic_score, critic_count, user_socre, user_count as ln_[original_variable_name].
vdgm$ln_Global_Sales = log(vdgm$Global_Sales)
vdgm$ln_Critic_Score = log(vdgm$Critic_Score)
vdgm$ln_Critic_Count = log(vdgm$Critic_Count)
vdgm$ln_User_Score = log(vdgm$User_Score)
vdgm$ln_User_Count = log(vdgm$User_Count)- Run a regression with the log of Y variable and report adjusted R-squared.
lm2 <- lm(ln_Global_Sales ~. - Name - ln_Critic_Score - ln_Critic_Count - ln_User_Score - ln_User_Count - Publisher - Developer - Global_Sales,data = vdgm)
summary(lm2)##
## Call:
## lm(formula = ln_Global_Sales ~ . - Name - ln_Critic_Score - ln_Critic_Count -
## ln_User_Score - ln_User_Count - Publisher - Developer - Global_Sales,
## data = vdgm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.4449 -0.6634 0.0008 0.6581 4.1020
##
## Coefficients: (26 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.173e+01 1.731e+01 -2.410 0.015989 *
## PlatformGBA 2.952e-02 1.019e-01 0.290 0.772038
## PlatformGC -1.457e-01 9.166e-02 -1.589 0.112065
## PlatformPC -1.713e+00 8.246e-02 -20.775 < 2e-16 ***
## PlatformPS2 4.156e-01 7.533e-02 5.517 3.64e-08 ***
## PlatformPS3 2.372e-01 7.770e-02 3.053 0.002281 **
## PlatformPSP -7.539e-02 8.301e-02 -0.908 0.363857
## PlatformWii 4.616e-01 7.830e-02 5.895 4.03e-09 ***
## PlatformX360 -1.543e-01 7.762e-02 -1.988 0.046896 *
## PlatformXB -4.391e-01 8.421e-02 -5.214 1.93e-07 ***
## GenreAdventure -4.542e-01 8.937e-02 -5.082 3.89e-07 ***
## GenreFighting 1.823e-02 7.654e-02 0.238 0.811773
## GenreMisc 1.828e-01 7.252e-02 2.520 0.011765 *
## GenrePlatform -3.026e-02 7.650e-02 -0.395 0.692495
## GenrePuzzle -5.231e-01 1.307e-01 -4.002 6.38e-05 ***
## GenreRacing -1.086e-01 6.522e-02 -1.665 0.095926 .
## GenreRole-Playing -2.975e-01 6.167e-02 -4.824 1.46e-06 ***
## GenreShooter -1.345e-01 5.630e-02 -2.389 0.016934 *
## GenreSimulation 2.685e-01 8.085e-02 3.321 0.000905 ***
## GenreSports -9.037e-02 6.348e-02 -1.423 0.154663
## GenreStrategy -5.684e-01 8.623e-02 -6.592 4.84e-11 ***
## RatingE10+ -2.287e-01 5.636e-02 -4.057 5.05e-05 ***
## RatingM -4.326e-01 6.253e-02 -6.919 5.19e-12 ***
## RatingT -3.404e-01 4.853e-02 -7.014 2.67e-12 ***
## Year_of_Release 1.925e-02 8.622e-03 2.233 0.025624 *
## Critic_Score 2.668e-02 1.659e-03 16.081 < 2e-16 ***
## Critic_Count 2.490e-02 1.196e-03 20.809 < 2e-16 ***
## User_Score -5.077e-02 1.465e-02 -3.464 0.000537 ***
## User_Count 5.051e-04 3.459e-05 14.605 < 2e-16 ***
## Action NA NA NA NA
## adventure NA NA NA NA
## fighting NA NA NA NA
## misc NA NA NA NA
## platform NA NA NA NA
## puzzle NA NA NA NA
## racing NA NA NA NA
## Role_playing NA NA NA NA
## Shooter NA NA NA NA
## simulation NA NA NA NA
## sports NA NA NA NA
## strategy NA NA NA NA
## DS NA NA NA NA
## GBA NA NA NA NA
## GC NA NA NA NA
## Pc NA NA NA NA
## Ps2 NA NA NA NA
## Ps3 NA NA NA NA
## PSP NA NA NA NA
## Wii NA NA NA NA
## X360 NA NA NA NA
## XB NA NA NA NA
## E NA NA NA NA
## `E10+` NA NA NA NA
## M NA NA NA NA
## T NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.026 on 4384 degrees of freedom
## Multiple R-squared: 0.4555, Adjusted R-squared: 0.452
## F-statistic: 131 on 28 and 4384 DF, p-value: < 2.2e-16- Run a regression with the log of Y variable as well as log of X variables generated in part d). Report adjusted R-squared.
lm3 <- lm(ln_Global_Sales ~. - Global_Sales - Name - Publisher - Developer - Critic_Score - Critic_Count -User_Score - ln_User_Count,data = vdgm)
summary(lm3)##
## Call:
## lm(formula = ln_Global_Sales ~ . - Global_Sales - Name - Publisher -
## Developer - Critic_Score - Critic_Count - User_Score - ln_User_Count,
## data = vdgm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3799 -0.6687 0.0071 0.6726 4.1881
##
## Coefficients: (26 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.970e+01 1.775e+01 -3.363 0.000779 ***
## PlatformGBA 4.049e-02 1.046e-01 0.387 0.698714
## PlatformGC -1.732e-01 9.380e-02 -1.847 0.064810 .
## PlatformPC -1.803e+00 8.390e-02 -21.490 < 2e-16 ***
## PlatformPS2 3.656e-01 7.701e-02 4.748 2.12e-06 ***
## PlatformPS3 2.341e-01 7.969e-02 2.937 0.003331 **
## PlatformPSP -1.171e-01 8.496e-02 -1.378 0.168325
## PlatformWii 4.492e-01 8.011e-02 5.607 2.19e-08 ***
## PlatformX360 -4.764e-02 7.862e-02 -0.606 0.544584
## PlatformXB -4.573e-01 8.619e-02 -5.305 1.18e-07 ***
## GenreAdventure -4.630e-01 9.146e-02 -5.062 4.32e-07 ***
## GenreFighting -6.030e-03 7.824e-02 -0.077 0.938575
## GenreMisc 2.037e-01 7.418e-02 2.746 0.006052 **
## GenrePlatform -1.965e-02 7.827e-02 -0.251 0.801784
## GenrePuzzle -4.934e-01 1.338e-01 -3.689 0.000228 ***
## GenreRacing -7.956e-02 6.672e-02 -1.192 0.233173
## GenreRole-Playing -3.034e-01 6.318e-02 -4.803 1.61e-06 ***
## GenreShooter -1.192e-01 5.761e-02 -2.069 0.038622 *
## GenreSimulation 2.675e-01 8.279e-02 3.231 0.001244 **
## GenreSports -9.166e-02 6.449e-02 -1.421 0.155272
## GenreStrategy -5.638e-01 8.829e-02 -6.386 1.88e-10 ***
## RatingE10+ -2.542e-01 5.765e-02 -4.409 1.06e-05 ***
## RatingM -4.071e-01 6.403e-02 -6.358 2.25e-10 ***
## RatingT -3.446e-01 4.975e-02 -6.928 4.89e-12 ***
## Year_of_Release 2.552e-02 8.847e-03 2.885 0.003936 **
## User_Count 6.437e-04 3.454e-05 18.640 < 2e-16 ***
## Action NA NA NA NA
## adventure NA NA NA NA
## fighting NA NA NA NA
## misc NA NA NA NA
## platform NA NA NA NA
## puzzle NA NA NA NA
## racing NA NA NA NA
## Role_playing NA NA NA NA
## Shooter NA NA NA NA
## simulation NA NA NA NA
## sports NA NA NA NA
## strategy NA NA NA NA
## DS NA NA NA NA
## GBA NA NA NA NA
## GC NA NA NA NA
## Pc NA NA NA NA
## Ps2 NA NA NA NA
## Ps3 NA NA NA NA
## PSP NA NA NA NA
## Wii NA NA NA NA
## X360 NA NA NA NA
## XB NA NA NA NA
## E NA NA NA NA
## `E10+` NA NA NA NA
## M NA NA NA NA
## T NA NA NA NA
## ln_Critic_Score 1.462e+00 9.917e-02 14.748 < 2e-16 ***
## ln_Critic_Count 5.554e-01 3.008e-02 18.461 < 2e-16 ***
## ln_User_Score -1.711e-01 8.012e-02 -2.136 0.032741 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.05 on 4384 degrees of freedom
## Multiple R-squared: 0.4299, Adjusted R-squared: 0.4262
## F-statistic: 118.1 on 28 and 4384 DF, p-value: < 2.2e-16- Which model (out of part c, e, and f) offers the highest adjusted R-squared? What would be the economic reasoning on why that particular model provides the best fit?
Ans: lm2 has the highest R-squared value. It makes sense that this has the best fit as the sales variable usually been increasing exponentially.
- Interpret the parameter estimates for each of ‘genre’ in plain English.
Ans: Simulation Games created the highest impact in terms of Global Sales and Sports created the lowest based on the parameter estimates.
- Interpret the parameter estimate for ‘rating’ in plain English.
Ans: Simulation Games created the highest impact in terms of Global Sales and Sports created the lowest based on the parameter estimates.
- Interpret the parameter estimate for ‘ln_user_count’ in plain English.
Ans: ln_User_Count has a p_value > 0.05. There we cannot reject the null hypothesis i.e. the coefficeint of ln_user_count = 0.